Wolfgang Pfeifer assigns German pusten, ultimately to "*b(e)u, *bh(e)u, *b(h)u "aufblasen, schwellen" (inflate, literally up blow; swell), along with Busen (bossom), and cetera, also suggesting "imitatitve". This is e.g. to blow out a candle.
I would add prusten, plustern, and perhaps compare to push, more aptly puff, but I am not sure about "imitative".
Wiktionary noted somewhere, that Pokorny's *bh(e)u- is not widely accepted.
There are uncountably many variations of this, what seems like an exercise in symbolic algebra, strewn accross the literature, with several different sounds and semantic interpretations. The both lists that you link to do include many such doublets, e.g. for "to speak". This is all very confusing.
If I might try to answer:
puh as a sound symbolic sign for the letting out of air in relieve does exist in German. French say boff in astonishment (German has comparable baff "astounded"). Donald Duck in the cartoons snores ra-puh, ra-puh. bah, pah, blergh as feigned spitting is common, too, not only in English. This is hardly indicative for the origin of the root, but might at least explain the authors line of reasoning, like, something like that, you know.
Your screenshot reads "146, 2. \bhes-, IE perd, to blow, breath". *perd is another supposed onomatopoea, *fart*. This seems improbable to relate to breath; I won't hold my breath for anything to come of it, but if there's a relation, it should be to smell, (of which Anatoli Liberman recently wrote a few fine blog posts, though ultimately inconclusive, linking smoke). German traditionally puns einen fahren lessen (to let one go), and if that links it to *per-, I'd further compare that to spühren, spirit, spirans, although spirit supposedly rather reflects *(s)peys- "to blow, breathe" [wiktionary, there also *peys-]. This looks as if it may relate to our *bhes-, with fortified *p through s-prefix (from privative, or reflexive *se?). On the other hand, as a kind of mythical, spiritual term, it compares loosely to *(s)pend- in the sense "to sprinkle" (cp. *pers- "to sprinkle", too). Therfore cp. further to spit, itterative sputter, perhaps Ger. putzen "brush" (e.g. brush teeth, though remember clean vs klein "small", cp. putus "boy" etc.), spray, indeed Ger. sprechen "speak", further spritzen, sprite. If I conflate all these, the task should be to disentangle them, but that's nigh impossible, if they had in fact become convolved, obscuring the origins (I agree here with @Aryaman's verdict above). Also compare pipe (whistle), Ger piepen (to squeek), but also poop, Ger. Pups (something small), to pop, beep, bloop.
There's a speculative hypothesis to be falsified, in seeing *b(h)e, *bhu, as an idiom more than a root. It would be appealing to think that: a) words have to come from somewhere, and sound imitation is a good fix-point for that; b) to breathe is a vivid symbol for being alive or just to be, variously reconstructed, e.g. *bhuH-, in the sense become, grow also comparable to "swell".
However, a) in my mind a noisy fricative with lip rounding and velarization immitates wind much better than a plosive, but see onomatopoetic piss, hiss, and consider especially shhhh for the sound of a stream; b) PIE is not the origin of speech per se; Scenarios in which recognizable onomatopoea were necessary are purely speculative; It might just as well come from much older layers, sound-immitative qualities merely reinforcing the idea, but not necessarily primary (except perhaps for language learning, I guess, but that position is not maintained in the literature, and concrete theories are not available to me). So, it might as well be deemed coincidence until further notice.
It would be great to have parallel evidence for the phonetic qualities of early *b(h), which does often derive /v/. Comparison to *w, eg. wind, Ger. wehen, to well or were, was (*Hwes-!) is apparently not really feasable.